Simple Search

Query:

Enter a query and click »Search«...

Format:

Display: entries per page entries

Zbl 0645.34007
Staněk, Svatoslav
A phase of the differential equation $y''=Q(t)y$ with a complex coefficient Q of the real variable. (English)
[J] Acta Univ. Palacki. Olomuc., Fac. Rerum Nat. 85, Math. 25, 57-75 (1986). ISSN 0231-9721

Let Q be a continuous complex-valued function on $j=(a,b)$ $(-D\le a<b\le \infty)$, Im Q(t)$\not\equiv 0$. Then there exist independent solutions $y\sb 1,y\sb 2$ of (Q): $y''=Q(t)y$ such that $y\sp 2\sb 1(t)+y\sp 2\sb 2(t)\ne 0$ for $t\in j$. We say that a $C\sp 3$ complex-valued function $\alpha$ is a (first) phase of (Q) if there exist independent solutions u, v of (Q), $u\sp 2(t)+v\sp 2(t)\ne 0$ on j, such that $\alpha '(t)=- w/(u\sp 2(t)+v\sp 2(t)),$ $t\in j$, where $w=uv'-u'v$. In the real case the phase of the second-order differential equation was introduced by {\it O. Boruvka} [Linear Differential Transformations of the Second Order (1971; Zbl 0222.34002)]. If $\alpha$ is a phase of (Q) then $b\sp{i\alpha (t)}/\sqrt{\alpha'(t)}$, $e\sp{-i\alpha (t)}/\sqrt{\alpha'(t)}$ are independent solutions of (Q). Phases of (Q) are used for the investigation of the decomposition of zeros of solutions to the differential equation (Q).
[S.Staněk]

MSC 2000:: *34A30 Linear ODE and systems
34M99 Differential equations in the complex domain

Keywords: phase; second-order differential equation; decomposition of zeros of solutions

Citations: Zbl 0222.34002

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Journal Journal

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Simple Search

Zentralblatt MATH Berlin [Germany]